Group #140366 — subgroups

Subgroups — #140366

Order 512.

254 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	#9594	C₂	1	2	yes	maximal
5	#7171	C₂	1	2	yes	maximal
6	C₂×#366	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	#1939	—	2	2	no	—
10	C₄×#107	C₂²	1	2	yes	—
11	C₂×#335	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1380	C₄	1	1	yes	characteristic
14	#1380	C₄	1	1	yes	characteristic
15	#1372	—	2	2	no	—
16	#1312	—	2	2	no	—
17	#1279	—	2	2	no	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×#107	C₂×C₄	1	2	yes	—
22	C₂×#107	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	4	2	no	—
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₄⋊C₄	—	2	2	no	—
29	#366	—	2	4	no	—
30	C₄×C₁₆	—	2	1	no	—
31	C₄×C₁₆	D₄	1	2	yes	—
32	C₄×C₁₆	Q₈	1	2	yes	—
33	C₄×C₁₆	Q₈	1	2	yes	—
34	#340	—	2	2	no	—
35	#336	—	4	2	no	—
36	#335	—	2	2	no	—
37	#335	—	2	2	no	—
38	C₈²	—	2	1	no	—
39	C₈²	—	2	1	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₂²×C₈	—	4	2	no	—
42	C₂²×C₈	D₈	1	2	yes	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	#41	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄²	—	4	2	no	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	4	2	no	—
53	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
54	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
55	#107	—	2	2	no	—
56	#107	—	2	2	no	—
57	#107	—	2	2	no	—
58	#107	M₁₆	1	4	yes	—
59	C₄⋊C₈	—	4	2	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	8	4	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	Q₁₆	1	2	yes	—
66	C₄×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₄×C₈	#41	1	1	yes	characteristic
70	#95	—	4	2	no	—
71	C₂²×C₄	—	4	2	no	—
72	C₂²×C₄	—	4	2	no	—
73	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
75	C₂²×C₄	#135	1	1	yes	characteristic
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	—	8	2	no	—
85	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₈	#125	1	1	yes	characteristic
89	C₂×C₈	C₂×M₁₆	1	2	yes	—
90	C₂×C₈	#131	1	2	yes	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄²	—	8	2	no	—
96	C₄²	—	8	2	no	—
97	C₄²	#121	1	1	yes	characteristic
98	C₄²	C₄⋊Q₈	1	1	yes	characteristic
99	C₄²	C₂×Q₁₆	1	1	yes	characteristic
100	C₄²	#135	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	—	2	2	no	—
103	C₁₆	—	4	2	no	—
104	C₂³	C₄×D₈	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	#389	1	1	yes	characteristic
113	C₂×C₄	#435	1	2	yes	—
114	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
115	C₂×C₄	#444	1	2	yes	—
116	C₂×C₄	#466	1	1	yes	characteristic
117	C₂×C₄	#467	1	1	yes	characteristic
118	C₂×C₄	#499	1	1	yes	characteristic
119	C₂×C₄	#500	1	1	yes	characteristic
120	C₂×C₄	#509	1	2	yes	—
121	C₂×C₄	#510	1	2	yes	—
122	C₈	—	2	4	no	—
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	4	2	no	—
126	C₈	—	16	2	no	—
127	C₈	#423	1	1	yes	characteristic
128	C₈	#424	1	1	yes	characteristic
129	C₈	#446	1	2	yes	—
130	C₂²	#1581	1	2	yes	—
131	C₂²	#1942	1	1	yes	characteristic
132	C₂²	#2176	1	1	yes	characteristic
133	C₂²	#2177	1	1	yes	characteristic
134	C₂²	#2178	1	2	yes	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	8	4	no	—
138	C₄	—	16	2	no	—
139	C₄	#1584	1	2	yes	—
140	C₄	#1940	1	1	yes	characteristic
141	C₄	#1946	1	1	yes	characteristic
142	C₄	#2220	1	2	yes	—
143	C₄	#2255	1	2	yes	—
144	C₂	#9598	1	1	yes	characteristic
145	C₂	#12073	1	2	yes	—
146	C₂	#12076	1	2	yes	—
147	C₂	#13168	1	1	yes	characteristic
148	C₂	#13169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#366

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

#9594

C₂

yes

maximal

#7171

C₂

yes

maximal

C₂×#366

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1939

—

C₄×#107

C₂²

yes

—

C₂×#335

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1380

C₄

yes

characteristic

#1380

C₄

yes

—

—

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂×#107

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

—

#366

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

Q₈

yes

—

C₄×C₁₆

Q₈

yes

—

—

—

—

—

C₈²

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

—

—

—

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#121

yes

characteristic

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

100

C₄²

#135

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₂³

C₄×D₈

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#389

yes

characteristic

113

C₂×C₄

#435

yes

—

114

C₂×C₄

C₄×Q₁₆

yes

characteristic

115

C₂×C₄

#444

yes

—

116

C₂×C₄

#466

yes

characteristic

117

C₂×C₄

#467

yes

characteristic

118

C₂×C₄

#499

yes

characteristic

119

C₂×C₄

#500

yes

characteristic

120

C₂×C₄

#509

yes

—

121

C₂×C₄

#510

yes

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

#423

yes

characteristic

128

C₈

#424

yes

characteristic

129

C₈

#446

yes

—

130

C₂²

#1581

yes

—

131

C₂²

#1942

yes

characteristic

132

C₂²

#2176

yes

characteristic

133

C₂²

#2177

yes

characteristic

134

C₂²

#2178

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#1584

yes

—

140

C₄

#1940

yes

characteristic

141

C₄

#1946

yes

characteristic

142

C₄

#2220

yes

—

143

C₄

#2255

yes

—

144

C₂

#9598

yes

characteristic

145

C₂

#12073

yes

—

146

C₂

#12076

yes

—

147

C₂

#13168

yes

characteristic

148

C₂

#13169

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₄×D₈

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#366	2
C₂×#1372	2
C₂×#1312	2
#9594	2
#7171	2

subgroup

index

Subgroups — #140366

Conjugacy classes of subgroups

Maximal subgroups